Some Properties of Kantorovich-Stancu-Type Generalization of Szász Operators including Brenke-Type Polynomials via Power Series Summability Method

Ilirias Research Institute, rr-Janina, No-2, Ferizaj 70000, Kosovo Department of Mathematics and Computer Sciences, University of Prishtina, Avenue Mother Teresa, No-5, Prishtine 10000, Kosovo Department of Mathematics, University of Haifa, 3498838 Haifa, Israel Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan


Introduction and Preliminaries
Let K ⊆ ℕ (set of natural numbers) and K m = fi ≤ m : i ∈ Kg. Then, the natural density or we can say asymptotic density of K is defined by σðKÞ = lim m ð1/mÞjK m j whenever the limit exists, where jK m j denotes the cardinality of the set K m . A sequence η = ðη i Þ is statistically convergent to L if for every ε > 0 and we write st − lim m η m = L. Let T = ðt nj Þ be a matrix and η = ðη j Þ be a sequence. The T − transform of the sequence η = ðη n Þ is defined by Tη = ðT n ðηÞÞ, ðTηÞ n = ∑ j t nj η j if the series converges for every n ∈ ℕ. We say that η is T − summable to the number L if ðTηÞ n converges to L. The summability matrix T is regular whenever lim j η j = L = lim n ðTηÞ n .
Let T = ðt nj Þ be a regular matrix. A sequence η = ðη j Þ is said to be T -statistically convergent (see [1]) to real number L if for any >0, lim n ∑ j:jη j −Lj≥ε t nj = 0, and we write st T − limη = L. If T is Cesàro matrix of order 1, then T -statistical convergence is reduced to the statistical convergence.
Let ðp j Þ be a sequence of real numbers such that p 0 > 0, p 1 , p 2 , ⋯ ≥ 0, and the corresponding power series pðuÞ = ∑ ∞ j=0 p j u j has radius of convergence R with 0 < R ≤ ∞. If lim u→R − ð1/pðuÞÞ∑ ∞ j=0 η j p j u j = L for all t ∈ ð0, RÞ, then we say that η = ðη j Þ is convergent in the sense of power series method (see [11,12]). Define AðuÞBðηuÞ = ∑ k≥0 p k ðηÞu k , where AðuÞ = ∑ j≥0 a j u j and BðuÞ = ∑ j≥0 b j u j are analytical functions such that a 0 ≠ 0 and b j ≠ 0 for all j ≥ 0 (see [13]).
Clearly, p k ðηÞ = ∑ k j=0 a k−j b j η j . Moreover, the power series method is regular if and only if lim u→R − ðp j u j /pðuÞÞ = 0 holds for each j ∈ f0, 1, ⋯g (see [14]).
We study a Korovkin-type theorem for the Kantorovich-Stancu-type Szász-Mirakyan operators via power series method. We determine the rate of convergence for these operators. Furthermore, we give a Voronovskaya-type theorem for T − statistical convergence. Such type of operators is widely studied by several authors (see [15][16][17][18][19]).

Journal of Function Spaces
Moreover, where Sðm, jÞ is the Stirling number of the second kind. Define a j = ðd j /dt j ÞAðtÞj t=1 and b j = ðd j /dt j ÞBðtÞj t=nx , for all j ≥ 0. Therefore, Theorem 2 with D = XL and Lemma 3 imply the following theorem.
where Sðm, ℓÞ is the Stirling number of the second kind.

Journal of Function Spaces
Thus, Lemma 1 completes the proof. By Theorem 6 and Lemma 3, we obtain the following result.
where Sðm, ℓÞ is the Stirling number of the second kind.

Main Results
We study here T − statistical convergence of the operators K α,β n . Note that the Korovkin-type theorem for T − statistical convergence was considered in [24] as follows: Theorem 11. Let ðB j Þ be a sequence of positive linear operators on C½0, 1 and let T = ðt nj Þ be a nonnegative regular summability matrix such that Then, for any f ∈ C½0, 1 where khk = max 0≤x≤1 jhðxÞj.
Based on the above theorem, we give the following result.
Proof. From Lemma 5, we have that st T − lim n kK α,β n e 0 − e 0 k = 0. Now, we will estimate the following expressions: Note that lim n→∞ ðB ′ ðnxÞÞ/ðBðnxÞÞ = 1. So from the last two relations we have that kK α,β n e 1 − e 1 k = 0. Moreover, Now proof follows directly from Theorem 11. This theorem is an extension of some known results for the Kantorovich-Stancu-type Szász-Mirakyan operators.

Journal of Function Spaces
Example 13 (see [6]). Under conditions given in Theorem 12, we define the following operators where the sequence ðx n Þ is given as follows: then By Theorem 11 we obtain st T − lim n kN n h − hk = 0, but the operators N n ðh, xÞ do not satisfy Theorem 12. Hence, the sequence ðN n Þ is not statistically convergent but it is T − statistically convergent. Remark 14. The sequence ðx n Þ is not statistically convergent and hence not convergent. As an example, consider the Cesáro matrix of order 2. where This proves that x = ðx n Þ is T − statistically convergent. We have N n e 0 , x ð Þ= 1 + x n ð Þ, By Example 13, this shows that N n ðh, xÞ does not satisfy Theorem 12.

Rate of Convergence
Modulus of continuity is defined by It is not hard to verify So, we can state the following.
Theorem 16. Let T = ðt ij Þ be a nonnegative regular summability matrix and h ∈ C½0, M ∩ E. If ðα n Þ is a sequence of positive real numbers such that ωðh, δ n Þ = st T − 0ðα n Þ, then 7 Journal of Function Spaces for any positive integer n: Proof. Let h ∈ C½0, M ∩ E: By positivity and linearity of K α,β n and (46), we see By applying the Cauchy-Schwartz inequality, we have Based on Examples 5 and Remark 8, we obtain By taking we get that kK α,β n h − hk ≤ 2 · ωðh, δ n Þ. Therefore, for every ε > 0, we have From the conditions that are given in the theorem, we have that kK α,β n h − hk = st T − 0ðα i Þ, as claimed. In the next result, we present the rate of convergence for the power summability method.
where the function ψ : ð0, MÞ ∩ E ⟶ ℝ is defined by relation which leads to If we set δ = ψðuÞ, then from the last inequality we have as required.

Voronovskaya-Type Theorems
First, we prove a Voronovskaya-type theorem for the operators under consideration.
for every x ∈ ½0, M.
Proof. Assume that h ′ , h ′ ′ ∈ C½0, M ∩ E and x ∈ ½0, M: By Taylor's formula, we have where ψðy − xÞ ⟶ 0 and y − x ⟶ 0. Applying in both sides of the above relation operators K α,β n , we obtain which implies Now, we will estimate this expression: Let ε > 0 and δ > 0 such that jψðy − xÞj < ε, where jy − xj < δ. We will split the above relation in two parts: On the other hand, from Proposition 9, condition (3), we get that jU 1 j ≤ ε 1 .
We extend the Voronovskaya-type theorem for T − statistical method for these operators. Consider operators N n from Example 13. We start with the following lemma.